## Binomial Theorem

Can you expand on ? I guess you would find that is quite easy to do. You can easily find that .

How about the expansion of . It is no longer easy.

It is no longer easy isn’t it. However, if we use Binomial Theorem, this expansion becomes an easy problem.

Binomial Theorem is a very intriguing topic in mathematics and it has wide range of applications.

## Theorem

Let  be real numbers (or complex, or polynomial). For any positive integer , we have:

where,

Proof:

We will use prove by induction. The base case  is obvious. Now suppose that the theorem is true for the case , that is assume that:

we will need to  show that, this is true for



Let us consider the left hand side of equation above



## We can now apply Pascal’s identity:



The equation above can be simplified to:

as we desired.

### Example 1:  Power rule in Calculus

In calculus, we always use the power rule that

We can prove this rule using Binomial Theorem.

Proof:

Recall that derivative for any continuous function f(x) is defined as:



Let  be a positive integer and let

The derivative of f(x) is:

### Example 2:  Binomial Distribution

Let X be the number of Head a sequence of n independent coin tossing. X is usually model by binomial distribution in probability model. Let  be the probability that a head show up in a toss, and let . The probability that there is  head in the sequence of  toss is:



We know that sum of all the probability must equal to 1. In order to show this, we can use Binomial Theorem. We have:



Please also check another article Gaussian Samples and N-gram language models ,Bayesian model , Monte Carlo for statistics knowledges.

## Monte Carlo Simulation

On a nice day 2 years ago, when I was on financial field. My boss sent our team an email. In this email, he would like to us propose some machine learning techniques to predict stock price.

So, after accepting the assignment from my manager, our team begin to research and apply some approaches for prediction. When we talk about Machine Learning, we often think of supervised and unsupervised learning. But one of the algorithms we applied is one that we usually forgotten  however equally highly effective algorithm: Monte Carlo Simulation.

## What is Monte Carlo simulation

Monte Carlo method is a technique that uses random numbers and probability to solve complex problems. The Monte Carlo simulation, or probability simulation, is a technique used to understand the impact of risk and uncertainty in financial sectors, project management, costs, and other forecasting machine learning models.[1]

Now let’s jump into python implementation to see how it applies,

## Python Implementation

In this task we used data of DXG stock dataset from 2017/01/01 to 2018/08/24 and we would like to know what is stock price after 10 days, 1 months and 3 months, respectively

We will simulate the return of stock and next price will be calculated by

P(t) = P(0) * (1+return_simulate(t))

Calculate mean and standard deviation of stock returns

miu = np.mean(stock_returns, axis=0)
dev = np.std(stock_returns)

Simulation process

simulation_df = pd.DataFrame()
last_price = init_price
for x in range(mc_rep):
count = 0
daily_vol = dev
price_series = []
price = last_price * (1 + np.random.normal(miu, daily_vol))
price_series.append(price)
for y in range(train_days):
if count == train_days-1:
break
price = price_series[count] * (1 + np.random.normal(miu, daily_vol))
price_series.append(price)
count += 1
simulation_df[x] = price_series

Visualization Monte Carlo Simulation

fig = plt.figure()
fig.suptitle('Monte Carlo Simulation')
plt.plot(simulation_df)
plt.axhline(y = last_price, color = 'r', linestyle = '-')
plt.xlabel('Day')
plt.ylabel('Price')
plt.show()

Now, let’s check with actual stock price after 10 days, 1 month and 3 months

plt.hist(simulation_df.iloc[9,:],bins=15,label ='histogram')
plt.axvline(x = test_simulate.iloc[10], color = 'r', linestyle = '-',label ='Price at 10th')
plt.legend()
plt.title('Histogram simulation and last price of 10th day')
plt.show()

We can see the most frequent occurrence price is pretty close with the actual price after 10th

If the forecast period is longer, the results is not good gradually

Simulation for next 1 month

After 3 months

## Conclusion

Monte Carlo simulation is used a lot in finance, although it has some weaknesses, but hopefully through this article, you will have a new look on the simulation application for forecasting.

## Reference

[1] Pratik Shukla, Roberto Iriondo, “Monte Carlo Simulation An In-depth Tutorial with Python”, medium, https://medium.com/towards-artificial-intelligence/monte-carlo-simulation-an-in-depth-tutorial-with-python-bcf6eb7856c8

Please also check Gaussian Samples and N-gram language models,
Bayesian Statistics  for statistics knowledges.

## Bayesian estimator of the Bernoulli parameter

In this post, I will explain how to calculate a Bayesian estimator. The taken example is very simple: estimate the parameter θ of a Bernoulli distribution.

A random variable X which has the Bernoulli distribution is defined as

with

In this case, we can write

.

In reality, the simplest way to eatimate θ is to sample X, count how many time the event occurs, then estimate the probability of occuring of event. This is exactly what the frequestists do.

In this post, I will show how do the Bayesian statisticians estimate θ. Although this doesn’t have a meaningful application, but it helps to understand how do the Bayesian statistics work. Let’s start.

## The posterior distribution of θ

Denote Y as the observation of the event. Given the parameter θ, if we sample the event n time, then the probability that the event occurs k time is (this is called the probability density function of Bernoulli )

In Bayesian statistics, we would like to calculate

$p(\theta|Y=y)$

By using the Bayesian formula, we have

With the prior distribution of theta as an Uniform distribution, p(θ) = 1, and it is easy to prove that

where Γ is the Gamma distribution. Hence, the posterior distribution is

Fortunately, this is the density function of the Beta distribution:

We use the following properties for evaluating the posterior mean and variance of theta.

If , then

## Simulation

In summary, the Bayesian estimator of theta is the Beta distribution with the  mean and variance as above. Here is the Python codes for simulating data and estimating theta

def bayes_estimator_bernoulli(data, a_prior=1, b_prior=1, alpha=0.05):
'''Input:
data is a numpy array with binary value, which has the distribution B(1,theta)    a_prior, b_prior: parameters of prior distribution Beta(a_prior, b_prior)    alpha: significant level of the posterior confidence interval for parameter    Model:
for estimating the parameter theta of a Bernoulli distribution    the prior distribution for theta is Beta(1,1)=Uniform[0,1]    Output:
a,b: two parameters of the posterior distribution Beta(a,b)
pos_mean: posterior estimation for the mean of theta
pos_var: posterior estimation for the var of theta'''
n = len(data)
k = sum(data)
a = k+1
b = n-k+1
pos_mean = 1.*a/(a+b)
pos_var = 1.*(a*b)/((a+b+1)*(a+b)**2)
## Posterior Confidence Interval
theta_inf, theta_sup = beta.interval(1-alpha,a,b)
print('Prior distribution: Beta(%3d, %3d)' %(a_prior,b_prior))
print('Number of trials: %d, number of successes: %d' %(n,k))
print('Posterior distribution: Beta(%3d,%3d)' %(a,b))
print('Posterior mean: %5.4f' %pos_mean)
print('Posterior variance: %5.8f' %pos_var)
print('Posterior std: %5.8f' %(np.sqrt(pos_var)))
print('Posterior Confidence Interval (%2.2f): [%5.4f, %5.4f]' %(1-alpha, theta_inf, theta_sup))
return a, b, pos_mean, pos_var

# Example
n = 129 # sample size
data = np.random.binomial(size=n, n=1, p=0.6)
a, b, pos_mean, pos_var = bayes_estimator_bernoulli(data)

And the result is

Prior distribution: Beta(  1,   1)
Number of trials: 129, number of successes: 76
Posterior distribution: Beta( 77, 54)
Posterior mean: 0.5878
Posterior variance: 0.00183556
Posterior std: 0.04284341
Posterior Confidence Interval (0.95): [0.5027, 0.6703]
In the simulation, we simulated 129 data from the Bernoulli distribution with θ=0.6. And the Bayesian estimation of θ is the posterior mean which is 0.5878.
This is a very simple example of Bayesian estimation. In reality, it is usually tricky to determinate a closed form solution of the posterior distribution from the given prior distribution. In that case, Monte Carlo technique is one of solutions to approximate the posterior distribution.
Please also check Gaussian Samples and N-gram language models for statistics knowledges.

## Background

In part 1 of my project, I built a unigram language model: it estimates the probability of each word in a text simply based on the fraction of times the word appears in that text.

The text used to train the unigram model is the book “A Game of Thrones” by George R. R. Martin (called train). The texts on which the model is evaluated are “A Clash of Kings” by the same author (called dev1), and “Gone with the Wind” — a book from a completely different author, genre, and time (called dev2).

In this part of the project, I will build higher n-gram models, from bigram (n=2) all the way to 5-gram (n=5). These models are different from the unigram model in part 1, as the context of earlier words is taken into account when estimating the probability of a word.

## Training the model

For a given n-gram model:

The example below shows the how to calculate the probability of a word in a trigram model:

## Dealing with words near the start of sentence

In higher n-gram language models, the words near the start of each sentence will not have a long enough context to apply the formula above. To make the formula consistent for those cases, we will pad these n-grams with sentence-starting symbols [S]. Below are two such examples under the trigram model:

From the above formulas, we see that the n-grams containing the starting symbols are just like any other n-gram. The only difference is that we count them only when they are at the start of a sentence. Lastly, the count of n-grams containing only [S] symbols is naturally the number of sentences in our training text:

## Dealing with unknown n-grams

Similar to the unigram model, the higher n-gram models will encounter n-grams in the evaluation text that never appeared in the training text. This can be solved by adding pseudo-counts to the n-grams in the numerator and/or denominator of the probability formula a.k.a. Laplace smoothing. However, as outlined part 1 of the project, Laplace smoothing is nothing but interpolating the n-gram model with a uniform model, the latter model assigns all n-grams the same probability:

Hence, for simplicity, for an n-gram that appears in the evaluation text but not the training text, we just assign zero probability to that n-gram. Later, we will smooth it with the uniform probability.

## Background

Language modeling — that is, predicting the probability of a word in a sentence — is a fundamental task in natural language processing. It is used in many NLP applications such as autocompletespelling correction, or text generation.

Currently, language models based on neural networks, especially transformers, are the state of the art: they predict very accurately a word in a sentence based on surrounding words. However, in this project, I will revisit the most classic of language model: the n-gram models.

## Data

In this project, my training data set — appropriately called train — is “A Game of Thrones”, the first book in the George R. R. Martin fantasy series that inspired the popular TV show of the same name.

Then, I will use two evaluating texts for our language model:

## What is a unigram?

In natural language processing, an n-gram is a sequence of n words. For example, “statistics” is a unigram (n = 1), “machine learning” is a bigram (n = 2), “natural language processing” is a trigram (n = 3), and so on. For longer n-grams, people just use their lengths to identify them, such as 4-gram, 5-gram, and so on. In this part of the project, we will focus only on language models based on unigrams i.e. single words.

Training the model

A language model estimates the probability of a word in a sentence, typically based on the the words that have come before it. For example, for the sentence “I have a dream”, our goal is to estimate the probability of each word in the sentence based on the previous words in the same sentence:

The unigram language model makes the following assumptions:

## Background

The goal of this project is to generate Gaussian samples in 2-D from uniform samples, the latter of which can be readily generated using built-in random number generators in most computer languages.

In part 1 of the project, the inverse transform sampling was used to convert each uniform sample into respective x and y coordinates of our Gaussian samples, which are themselves independent standard normal (having mean of 0 and standard deviation of 1):

However, this method uses the inverse cumulative distribution function (CDF) of the Gaussian distribution, which is not well-defined. Therefore, we approximated this function using the simple Taylor series. However, this only samples accurately near the Gaussian mean, and under-samples more extreme values at both ends of the distribution.

In part 2 of the project, we used the Box-Muller transform, a more direct method to transform the uniform samples into Gaussian ones. The implementation of the algorithm is quite simple, as seen below, but its derivation requires some clever change of variables: instead of sampling Gaussian x and y coordinates for each point, we will sample a uniformly-distributed angle (from 0 to 2π) and an exponentially-distributed random variable that represents half of squared distance of the sample to the origin.

In this part of the project, I will present an even simpler method than the above two methods. Even better, this method is one that every statistics students are already familiar with.

## The Central Limit Theorem

It turns out, we can rely on the most fundamental principle of statistics to help us generate Gaussian samples: the central limit theorem. In very simple terms, the central limit theorem states that:

Given n independent random samples from the same distribution, their sum will converge to a Gaussian distribution as n gets large.

Therefore, to generate a Gaussian sample, we can just generate many independent uniform samples and add them together! We then repeat this routine to generate the next standard Gaussian sample until we get enough samples for our x-coordinates. Finally, we just repeat the same steps to generate the y coordinates.

Note that this method will work even if the samples that we start with are not uniform — they are Poisson-distributed, for example. This is because the central limit theorem holds for virtually all probability distributions *cough let’s not talk about the Cauchy distribution cough*.

## Generate Gaussian samples by central limit theorem

To generate, say, 1000 Gaussian sums (n_sums = 1000) where each is the sum of 100 uniform samples (n_additions = 100):

n_points = 1000# 0. Initialize random number generator
rng = np.random.RandomState(seed=24) # 1. Generate matrix of uniform samples
uniform_matrix = rng.uniform(size=(n_additions, n_points))# 2. Sum uniform elements down each column to get all Gaussian sums
gaussians = uniform_matrix.sum(axis=0)

We can apply the above method to generate Gaussian samples for each coordinate, using different random number generator for x and for y to ensure that the coordinates are independent from each other. Visualizing the intermediate sums after each addition, we see that:

## Background

In part 1 of this project, I’ve shown how to generate Gaussian samples using the common technique of inversion sampling:

The bad news: the Gaussian inverse CDF is not well-defined, so we have to approximate that function, and the simple Taylor series was used. However, this only samples accurately near the Gaussian mean, and under-samples more extreme values at both ends of the distribution.

Therefore, in this part of the project, we will investigate a more “direct” sampling method that does not depend on the approximation of the Gaussian inverse CDF. This method is called the Box-Muller transform, after the two mathematicians who invented the method in 1958: the British George E. P. Box and the American Mervin E. Muller.

## How does the Box-Muller transform work?

For this project, my goal is to generate Gaussian samples in two dimensions i.e. generating samples whose x and y coordinates are independent standard normals (Gaussian with zero mean and standard deviation of 1). In part 1, I used the inverse Gaussian CDF to generate separately the x and y coordinates from their respective uniform samples (U₁ and U₂):

For the Box-Muller transform, I will also start with the same two uniform samples. However, I will transform these uniform samples into the x and y coordinates using much simpler formulas:

Despite the strong coupling between U₁ and U₂ in each of the two formulas above, the generated x and y coordinates, which are both standard Gaussians, are still surprisingly independent from each other! In the derivation of the Box-Muller transform that follows, I will demonstrate why this is indeed this case.

## Derivation of Box-Muller transform

We know that for any two independent random variables x and y, the joint probability density f(x, y) is simply the product of the individual density functions: f(x) and f(y). Furthermore, Pythagorus theorem allows us to combine the x² and y² term in each of the Gaussian density function. This results in the -r²/2 term in the exponential of the joint distribution, where r is the distance from the origin to the 2-D Gaussian sample.

To make it simpler, we then define a variable s that is equal to r²/2. In other words, s is simply the half of squared distance from the origin of our Gaussian sample. Written this way, the joint PDF is simply the product of a constant (1/2π) and an exponential (e⁻ˢ).

## Background

Gaussian sampling — that is, generating samples from a Gaussian distribution — plays an important role in many cutting-edge fields of data science, such as Gaussian processvariational autoencoder, or generative adversarial network. As a result, you often see functions like tf.random.normal in their tutorials.

But, deep down, how does a computer know how to generate Gaussian samples? This series of blog posts will show 3 different ways that we can program our computer (via Python) to do so. You will also see how R and Python generate Gaussian samples using modified versions of some of these methods.

## Starting point: the uniform number generator

Of course, we can’t generate Gaussian samples from thin air. Instead, we start with a random number generator that exists in almost all programming languages: the uniform random number generator. It generates a random number that could take any value between 0 and 1. For Python, the numpy.random module uses the Mersenne twister to generate a uniformly-distributed float that is in the interval [0.0, 1.0).

Since Gaussians are better visualized in 2 dimensions — we are all familiar with the Gaussian “blob” in the xy-plane — I will demonstrate the 3 sampling methods in 2-D, especially since one of the methods do in fact generate Gaussians in two dimensions at the same time.

As a result, this series is broken down into 3 parts (see accompanying image):

## Method 1: Inverse transform sampling

This is the most basic, yet most common, way to convert a uniform random sample into a random sample of any distribution, including Gaussian. This method works by applying the inverse function of the Gaussian CDF (cumulative distribution function) to transform a uniform sample to a Gaussian sample.

To make sure that the Gaussian samples for the x- and y-coordinates are independent, we can use two different uniform samples, one for x (U₁), and one for y (U₂). These two uniform samples can be generated using two different random number generators (two different RandomState initialized by different seeds, for example) so that they are independent in the first place.

## How does this work?

This method works by exploiting a mind-blowing principle:

For any distribution, the cumulative probability is always uniformly distributed.

The arithmetic proof of this principle is straight-forward but rather boring, and you can view it from this lecture. Instead, I will show the geometric interpretation of this principle. But first, let’s clarify what cumulative probability is:

## Background

In previous parts of my project, I built different n-gram models to predict the probability of each word in a given text. This probability is estimated using an n-gram — a sequence of word of length n — which contains the word. The below formula shows how the probability of the word “dream” is estimated as part of the trigram “have a dream”:

We train the n-gram models on the book “A Game of Thrones” by George R. R. Martin (called ). We then evaluate the models on two texts: “A Clash of Kings” by the same author (called ), and “Gone with the Wind” — a book from a completely different author, genre, and time (called ).

The metric to evaluate the language model is average log likelihood: the average of the log probability that the model assigns to each word in the evaluation text.

Often, log of base 2 is applied to each probability, as is the case in the first two parts of the project. Nevertheless, in this part, I will use natural log, as it makes it simpler to derive the formulas that we will be using.

## Problem

In part 2, the various n-gram models — from unigram to 5-gram — were evaluated on the evaluation texts ( and , see graphs below).

From this, we notice that:

• Bigram model perform slightly better than unigram model. This is because the previous word to the bigram can provide important context to predict the probability of the next word.
• Surprisingly, trigram model and up are much worse than bigram or unigram models. This is largely due to the high number of trigrams, 4-grams, and 5-grams that appear in the evaluation texts but nowhere in the training text. Hence, their predicted probability is zero.
• For most n-gram models, their performance is slightly improved when we interpolate their predicted probabilities with the uniform model. This seems rather counter-intuitive, since the uniform model simply assigns equal probability to every word. However, as explained in part 1, interpolating with this “dumb” model reduces the overfit and variance of the n-gram models, helping them generalize better to the evaluation texts.

In this part of the project, we can extend model interpolation even further: instead of separately combining each n-gram model with the uniform, we can interpolate different n-gram models with one another, along with the uniform model.

## What to interpolate?

The first question to ask when interpolating multiple models together is:

To answer this question, we use the simple strategy outlined below:

1. First, we start with the uniform model. This model will have very low average log likelihoods on the evaluation texts, since it assigns every word in the text the same probability.
2. Next, we interpolate this uniform model with the unigram model and re-evaluate it on the evaluation texts. We naively assume that the models will have equal contribution to the interpolated model. As a result, each model will have the same interpolation weight of 1/2.
3. We then add the bigram model to the mix. Similarly, in this 3-model interpolation, each model will simply have the same interpolation weight of 1/3.
4. We keep adding higher n-gram models to the mix, while keeping the mixture weights the same across models, until we reach the 5-gram model. After each addition, the combined model will be evaluated against the two evaluation texts,  and .

## Coding the interpolation

In part 2, each evaluation text had a corresponding probability matrix. This matrix has 6 columns — one for each model — and each row of the matrix represents the probability estimates of each word under the 6 models. However, since we want to optimize the model performance on both evaluation texts, we will vertically concatenate these probability matrices into one big evaluation probability matrix (803176 rows × 6 columns):